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Isothermal–isobaric ensemble - Wikipedia

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The isothermal–isobaric ensemble (constant temperature and constant pressure ensemble) is a statistical mechanical ensemble that maintains constant temperature $T\,$ and constant pressure $P\,$ applied. It is also called the $NpT$ -ensemble, where the number of particles $N\,$ is also kept as a constant. This ensemble plays an important role in chemistry as chemical reactions are usually carried out under constant pressure condition.^[1] The NPT ensemble is also useful for measuring the equation of state of model systems whose virial expansion for pressure cannot be evaluated, or systems near first-order phase transitions.^[2]

In the ensemble, the probability of a microstate $i$ is $Z^{-1}e^{-\beta (E(i)+pV(i))}$ , where $Z$ is the partition function, $E(i)$ is the internal energy of the system in microstate $i$ , and $V(i)$ is the volume of the system in microstate $i$ .

The probability of a macrostate is $Z^{-1}e^{-\beta (E+pV-TS)}=Z^{-1}e^{-\beta G}$ , where $G$ is the Gibbs free energy.

Derivation of key properties

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The partition function for the $NpT$ -ensemble can be derived from statistical mechanics by beginning with a system of $N$ identical atoms described by a Hamiltonian of the form $\mathbf {p} ^{2}/2m+U(\mathbf {r} ^{n})$ and contained within a box of volume $V=L^{3}$ . This system is described by the partition function of the canonical ensemble in 3 dimensions:

$Z^{sys}(N,V,T)={\frac {1}{\Lambda ^{3N}N!}}\int _{0}^{L}...\int _{0}^{L}d\mathbf {r} ^{N}\exp(-\beta U(\mathbf {r} ^{N}))$ ,

where $\Lambda ={\sqrt {h^{2}\beta /(2\pi m)}}$ , the thermal de Broglie wavelength ( $\beta =1/k_{B}T\,$ and $k_{B}\,$ is the Boltzmann constant), and the factor $1/N!$ (which accounts for indistinguishability of particles) both ensure normalization of entropy in the quasi-classical limit.^[2] It is convenient to adopt a new set of coordinates defined by $L\mathbf {s} _{i}=\mathbf {r} _{i}$ such that the partition function becomes

$Z^{sys}(N,V,T)={\frac {V^{N}}{\Lambda ^{3N}N!}}\int _{0}^{1}...\int _{0}^{1}d\mathbf {s} ^{N}\exp(-\beta U(\mathbf {s} ^{N}))$ .

If this system is then brought into contact with a bath of volume $V_{0}$ at constant temperature and pressure containing an ideal gas with total particle number $M$ such that $M-N\gg N$ , the partition function of the whole system is simply the product of the partition functions of the subsystems:

$Z^{sys+bath}(N,V,T)={\frac {V^{N}(V_{0}-V)^{M-N}}{\Lambda ^{3M}N!(M-N)!}}\int d\mathbf {s} ^{M-N}\int d\mathbf {s} ^{N}\exp(-\beta U(\mathbf {s} ^{N}))$ .

{\displaystyle V} — The system (volume $V$ ) is immersed in a much larger bath of constant temperature, and closed off such that particle number remains fixed. The system is separated from the bath by a piston that is free to move, such that its volume can change.

The integral over the $\mathbf {s} ^{M-N}$ coordinates is simply $1$ . In the limit that $V_{0}\rightarrow \infty$ , $M\rightarrow \infty$ while $(M-N)/V_{0}=\rho$ stays constant, a change in volume of the system under study will not change the pressure $p$ of the whole system. Taking $V/V_{0}\rightarrow 0$ allows for the approximation $(V_{0}-V)^{M-N}=V_{0}^{M-N}(1-V/V_{0})^{M-N}\approx V_{0}^{M-N}\exp(-(M-N)V/V_{0})$ . For an ideal gas, $(M-N)/V_{0}=\rho =\beta P$ gives a relationship between density and pressure. Substituting this into the above expression for the partition function, multiplying by a factor $\beta P$ (see below for justification for this step), and integrating over the volume V then gives

$\Delta ^{sys+bath}(N,P,T)={\frac {\beta PV_{0}^{M-N}}{\Lambda ^{3M}N!(M-N)!}}\int dVV^{N}\exp({-\beta PV})\int d\mathbf {s} ^{N}\exp(-\beta U(\mathbf {s} ))$ .

The partition function for the bath is simply $\Delta ^{bath}=V_{0}^{M-N}/[(M-N)!\Lambda ^{3(M-N)}$ . Separating this term out of the overall expression gives the partition function for the $NpT$ -ensemble:

$\Delta ^{sys}(N,P,T)={\frac {\beta P}{\Lambda ^{3N}N!}}\int dVV^{N}\exp(-\beta PV)\int d\mathbf {s} ^{N}\exp(-\beta U(\mathbf {s} ))$ .

Using the above definition of $Z^{sys}(N,V,T)$ , the partition function can be rewritten as

$\Delta ^{sys}(N,P,T)=\beta P\int dV\exp(-\beta PV)Z^{sys}(N,V,T)$ ,

which can be written more generally as a weighted sum over the partition function for the canonical ensemble

$\Delta (N,P,T)=\int Z(N,V,T)\exp(-\beta PV)CdV.\,\;$

The quantity $C$ is simply some constant with units of inverse volume, which is necessary to make the integral dimensionless. In this case, $C=\beta P$ , but in general it can take on multiple values. The ambiguity in its choice stems from the fact that volume is not a quantity that can be counted (unlike e.g. the number of particles), and so there is no “natural metric” for the final volume integration performed in the above derivation.^[2] This problem has been addressed in multiple ways by various authors,^[3]^[4] leading to values for C with the same units of inverse volume. The differences vanish (i.e. the choice of $C$ becomes arbitrary) in the thermodynamic limit, where the number of particles goes to infinity.^[5]

The $NpT$ -ensemble can also be viewed as a special case of the Gibbs canonical ensemble, in which the macrostates of the system are defined according to external temperature $T$ and external forces acting on the system $\mathbf {J}$ . Consider such a system containing $N$ particles. The Hamiltonian of the system is then given by ${\mathcal {H}}-\mathbf {J} \cdot \mathbf {x}$ where ${\mathcal {H}}$ is the system's Hamiltonian in the absence of external forces and $\mathbf {x}$ are the conjugate variables of $\mathbf {J}$ . The microstates $\mu$ of the system then occur with probability defined by ^[6]

$p(\mu ,\mathbf {x} )=\exp[-\beta {\mathcal {H}}(\mu )+\beta \mathbf {J} \cdot \mathbf {x} ]/{\mathcal {Z}}$

where the normalization factor ${\mathcal {Z}}$ is defined by

${\mathcal {Z}}(N,\mathbf {J} ,T)=\sum _{\mu ,\mathbf {x} }\exp[\beta \mathbf {J} \cdot \mathbf {x} -\beta {\mathcal {H}}(\mu )]$ .

This distribution is called generalized Boltzmann distribution by some authors.^[7]

The $NpT$ -ensemble can be found by taking $\mathbf {J} =-P$ and $\mathbf {x} =V$ . Then the normalization factor becomes

${\mathcal {Z}}(N,\mathbf {J} ,T)=\sum _{\mu ,\{\mathbf {r} _{i}\}\in V}\exp[-\beta PV-\beta (\mathbf {p} ^{2}/2m+U(\mathbf {r} ^{N}))]$ ,

where the Hamiltonian has been written in terms of the particle momenta $\mathbf {p} _{i}$ and positions $\mathbf {r} _{i}$ . This sum can be taken to an integral over both $V$ and the microstates $\mu$ . The measure for the latter integral is the standard measure of phase space for identical particles: ${\textrm {d}}\Gamma _{N}={\frac {1}{h^{3}N!}}\prod _{i=1}^{N}d^{3}\mathbf {p} _{i}d^{3}\mathbf {r} _{i}$ .^[6] The integral over $\exp(-\beta \mathbf {p} ^{2}/2m)$ term is a Gaussian integral, and can be evaluated explicitly as

$\int \prod _{i=1}^{N}{\frac {d^{3}\mathbf {p} _{i}}{h^{3}}}\exp {\bigg [}-\beta \sum _{i=1}^{N}{\frac {p_{i}^{2}}{2m}}{\bigg ]}={\frac {1}{\Lambda ^{3N}}}$ .

Inserting this result into ${\mathcal {Z}}(N,P,T)$ gives a familiar expression:

${\mathcal {Z}}(N,P,T)={\frac {1}{\Lambda ^{3N}N!}}\int dV\exp(-\beta PV)\int d\mathbf {r} ^{N}\exp(-\beta U(\mathbf {r} ))=\int dV\exp(-\beta PV)Z(N,V,T)$ .^[6]

This is almost the partition function for the $NpT$ -ensemble, but it has units of volume, an unavoidable consequence of taking the above sum over volumes into an integral. Restoring the constant $C$ yields the proper result for $\Delta (N,P,T)$ .

From the preceding analysis it is clear that the characteristic state function of this ensemble is the Gibbs free energy,

$G(N,P,T)=-k_{B}T\ln \Delta (N,P,T)\;\,$

This thermodynamic potential is related to the Helmholtz free energy (logarithm of the canonical partition function), $F\,$ , in the following way:^[1]

$G=F+PV.\;\,$

^ ^a ^b Dill, Ken A.; Bromberg, Sarina; Stigter, Dirk (2003). Molecular Driving Forces. New York: Garland Science.
^ ^a ^b ^c ^d ^e Frenkel, Daan.; Smit, Berend (2002). Understanding Molecular Simluation. New York: Academic Press.
^ Attard, Phil (1995). "On the density of volume states in the isobaric ensemble". Journal of Chemical Physics. 103 (24): 9884–9885. Bibcode:1995JChPh.103.9884A. doi:10.1063/1.469956.
^ Koper, Ger J. M.; Reiss, Howard (1996). "Length Scale for the Constant Pressure Ensemble: Application to Small Systems and Relation to Einstein Fluctuation Theory". Journal of Physical Chemistry. 100 (1): 422–432. doi:10.1021/jp951819f.
^ Hill, Terrence (1987). Statistical Mechanics: Principles and Selected Applications. New York: Dover.
^ ^a ^b ^c Kardar, Mehran (2007). Statistical Physics of Particles. New York: Cambridge University Press.
^ Gao, Xiang; Gallicchio, Emilio; Roitberg, Adrian (2019). "The generalized Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy". The Journal of Chemical Physics. 151 (3): 034113. arXiv:1903.02121. Bibcode:2019JChPh.151c4113G. doi:10.1063/1.5111333. PMID 31325924. S2CID 118981017.
^ McDonald, I. R. (1972). " $NpT$ -ensemble Monte Carlo calculations for binary liquid mixtures". Molecular Physics. 23 (1): 41–58. Bibcode:1972MolPh..23...41M. doi:10.1080/00268977200100031.
^ Wood, W. W. (1970). " $NpT$ -Ensemble Monte Carlo Calculations for the Hard Disk Fluid". Journal of Chemical Physics. 52 (2): 729–741. Bibcode:1970JChPh..52..729W. doi:10.1063/1.1673047.
^ Schmidt, Jochen; VandeVondele, Joost; Kuo, I. F. William; Sebastiani, Daniel; Siepmann, J. Ilja; Hutter, Jürg; Mundy, Christopher J. (2009). "Isobaric-Isothermal Molecular Dynamics Simulations Utilizing Density Functional Theory:An Assessment of the Structure and Density of Water at Near-Ambient Conditions". Journal of Physical Chemistry B. 113 (35): 11959–11964. doi:10.1021/jp901990u. OSTI 980890. PMID 19663399.